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**Conception et modélisation d'un montage de fabrication pour **

le balançage optimisé d'une famille de pièces Sajid Ullah BUTT Jury M. Cornel Mihai NICOLESCU, Professeur, KTH, Stockholm, Sweden Rapporteur M. Jean-François RIGAL, Professeur, LAMCOS, INSA Lyon, France Rapporteur M. Henri PARIS, Professeur, G.SCOP, Université Joseph Fourier, Grenoble, France Examinateur M. Jean-François ANTOINE, Maitre de conférences, IUT de Nancy Brabois, France Co-directeur de thèse M. Patrick MARTIN, Professeur, LCFC, Arts et Métiers ParisTech, Metz, France Directeur de thèse Arts et Métiers ParisTech - Centre de Metz Laboratoire de Conception Fabrication Commande EA 4495

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**Conclusion and Perspectives**

Presentation layout Context Kinematic Model Mechanical Model Conclusion and Perspectives Context Positioning errors Compensation Objectives Workpiece repositioning through locators All the elements rigid Proposed fixturing system Analytical formulation Large displacements and Homogeneous Transformation Matrices Deformation of elastic elements and rigid body displacement of part on fixture under load Deformation of locators and contacts Lagrangian formulation Negligible friction, Small Displacements Convergence of non-linear contact deformation Work realized Conclusion Future work Case Study Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Conclusion and Perspectives**

Presentation layout Context Kinematic Model Mechanical model Conclusion Context Kinematic Model Mechanical Model Conclusion and Perspectives Context Positioning errors Compensation Objectives Workpiece repositioning through locators All the elements rigid Proposed fixturing system Analytical formulation Large displacements and Homogeneous Transformation Matrices Deformation of elastic elements and rigid body displacement of part on fixture under load Deformation of locators and contacts Lagrangian formulation Negligible friction, Small Displacements Convergence of non-linear contact deformation Work realized Conclusion Future work Case Study Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Context Context Kinematic Model Mechanical model Conclusion Optimal balancing The final product should have a minimum allowance for better machining In case of perfect positioning, minimum rough part radius should have to be r + h ∆ is the positioning error between the final product and the rough part’s central axis The minimum radius of the rough part has to be R for a good machining operation More positioning error will increase the material waste R Initial surface r Final part ∆ Allowance > Min chip thickness h r h ∆ L Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Context Context Kinematic Model Mechanical model Conclusion Workpiece/machine tool Positioning error Variation among the parts of the same part family cause the positioning error during fixturing Positioning error of the workpiece affects the quality of the final product Column Spindle Tool Kinematics defects Deformation due to forces Tool wear Effect of heat NC Code errors Part Locators placement Pallet Geometric/form defects Base Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Possible placement of locators**

Positioning errors Context Kinematic Model Mechanical model Conclusion Y -Y Z -X X -Z Placement of locators Block the 6-DOFs of the part Placement procedure Choose the locating surfaces taking into account the constraints of accessibility, load, external force and movements (Somashekar 2002) Select the locators configurations (3-2-1, 3-2-1C, etc.) Choose the locators positions for the part stability (Roy & Liao, 2002; Zirmi et al. 2009) 3-2-1 3-2-1C 4-1-1 (H. Paris, 1995) Possible placement of locators Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Positioning errors Context Kinematic Model Mechanical model Conclusion Geometrical and form defects When workpiece is placed directly on the locators Local geometrical defects cause the orientation error The orientation error have more effect on the final product quality than the translation error (Asante, 2009) Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Positioning errors Deformation of locators under external load**

Context Kinematic Model Mechanical model Conclusion Deformation of locators under external load The locators and their contacts deform under clamping and machining forces Deformation depends upon the stiffness of the locators Hertz contact theory may be applied to calculate the contact deformation Locators deformations induce the workpiece displacement F Including contact deformation F F Zero contact deformation F Z Y X Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Positioning errors How to Compensate these errors?**

Context Kinematic Model Mechanical model Conclusion Machine tool/kinematic chain defects Machine tool position uncertainty Kinematic chain Kinematic defects increase with the increase the number of machine axes Other Defects Defects due to heat generation NC code defects Tool wear How to Compensate these errors? Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Error compensation Context Kinematic Model Mechanical model Conclusion Existing methods Changing the part program Easiest way (Ramesh et al. 2000) Orientation of the machine tool Disadvantages Need 4 or 5 axis machines Very expensive for the existing production line Column Tool Tool orientation Part program Part Part Actual position Ideal position Compensated position Pallet Base (Zhu et al. 2012) Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Error compensation Our proposal**

Context Kinematic Model Mechanical model Conclusion Our proposal A 6-DOF workpiece repositioning system is proposed A baseplate is introduced to avoid the positioning error caused by the geometrical defects Repositioning is performed through the positioning of the 6 locators Column Tool Part Baseplate Part Baseplate Part 6 DOF repositioning Pallet Base

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**Objective Develop a fixturing system which can Measure Calculate**

Context Kinematic Model Mechanical model Conclusion Develop a fixturing system which can Hold custom single and complex parts Perform 6-DOF Repositioning of the part at desired position Added to Single machine unit or production/assembly line Minimum modifications on existing production line Measure Calculate Compensate Base Column Pallet Tool Part Baseplate Conveyor Previous Workstation Reconfigurable Pallet Single machine unit Production/Assembly line Part Baseplate Part Baseplate Part Baseplate Reconfigurable Pallet Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Objective Context Kinematic Model Mechanical model Conclusion Error compensation principle Measure Workpiece geometrical errors Offline: using CMM Online: Integrated sensors Calculate The advancement of locators required to compensate the errors using Homogeneous Transformation Matrices and Large displacements (Kinematic model) The errors due to deformation of elastic elements under load using Small Displacement hypothesis (Mechanical model) Compensate Through the axial advancement of 6 locators Geometrical errors Mechanical errors Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Conclusion and Perspectives**

Presentation layout Context Kinematic Model Mechanical model Conclusion Context Kinematic Model Mechanical Model Conclusion and Perspectives Context Positioning errors Compensation Objectives Workpiece repositioning through locators All the elements rigid Proposed fixturing system Analytical formulation Large displacements and Homogeneous Transformation Matrices Deformation of elastic elements and rigid body displacement of part on fixture under load Deformation of locators and contacts Lagrangian formulation Negligible friction, Small Displacements Convergence of non-linear contact deformation Work realized Conclusion Future work Case Study Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Kinematic model Context Kinematic Model Mechanical model Conclusion Objective Repositioning of a part of a part family placed roughly with a precision of some millimeters Components Part (Hip prosthesis) Baseplate (Cuboid) 6-Locators (Axial movement) Pallet All the elements are rigid X Z Y O (Machine/Pallet reference) P 6 4 5 2 3 1 Baseplate part Part Baseplate Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Formulation [PPF] XF XP [Pb’F]=[PbP] [PbP] Correction Xb’ Xb [PPF ]**

Context Kinematic Model Mechanical model Conclusion Error to be corrected [PPF] XF XP Baseplate Surface normals [Pb’F]=[PbP] [PPF] = [Pbb’] [PbP] Xb’ Xb Rigid link Correction [PPF ] [POb’] [POb] XP ZP YP part Y3 Z3 X3 b XO P Baseplate Initial baseplate placement on the locators 5 Baseplate correction through locators 6 4 2 X Z Y 1 3 O (Machine/Pallet reference) Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Initial Position of the workpiece (2D Simulation) Context Kinematic Model Mechanical model Conclusion Point P (intersection of two centerlines) Definition of a plane Simulation in 2D CPT 12/14 Hip Prosthesis Zimmer Neck Stem Y RMS P Min Material (Chebyshev) X Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Initial Position of the workpiece (2D Simulation) Context Kinematic Model Mechanical model Conclusion Point P (intersection of two centerlines) Definition of a plane Simulation in 2D Neck Stem Y RMS P Min Material (Chebyshev) X Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Compensation of errors Context Kinematic Model Mechanical model Conclusion Final calculated Position 2D Schematic explanation Calculating point of contact in axis 1* 1’ 2* 2’ 1 2 Z Position calculation in 3D X Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Simulation procedure: CAD Modeling Context Kinematic Model Mechanical model Conclusion Boolean Operation Inverse impression of the workpiece for the simulation of cutting tool path D=3mm Cavity with original workpiece dimensions Baseplate Pallet Workpiece Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Case study Context Kinematic Model Mechanical model Conclusion Initial Data Final required part position Gray: Machined surface Orange: Rough surface Calculated positions of locators Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Case study Context Kinematic Model Mechanical model Conclusion Calculated positions of locators Simulated machining Positioning error of workpiece after correction Positioning error of workpiece after second side correction Calculated positions of locators Final Product Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Robustness of model/Sensitivity Analysis**

Context Kinematic Model Mechanical model Conclusion Use of Plucker matrix Precision of workpiece displacement as a function of locators’ positioning precision Position uncertainty = Geometrical uncertainty + uncertainty due to temperature Precision of locator Worst case Sensitivity analysis Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Conclusion and Perspectives**

Presentation layout Context Kinematic Model Mechanical model Conclusion Context Kinematic Model Mechanical Model Conclusion and Perspectives Context Positioning errors Compensation Objectives Workpiece repositioning through locators All the elements rigid Proposed fixturing system Analytical formulation Large displacements and Homogeneous Transformation Matrices Deformation of elastic elements and rigid body displacement of part on fixture under load Deformation of locators and contacts Lagrangian formulation Negligible friction, Small Displacements Convergence of non-linear contact deformation Work realized Conclusion Future work Case Study Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Mechanical model Context Kinematic Model Mechanical model Conclusion Clamping and machining forces and moments Elements designed to be rigid Workpiece baseplate assembly Mass elements Elastic elements Locators (body and contact) Baseplate at contacts Clamps with imposed external displacements Small displacement hypothesis Friction neglected Effects of heat neglected No slippage of clamps at contact {XE}2 Z X Y P [K]1 [K]2 [K]3 [K]4 [K]5 [K]6 {XE}1 [KE]2 [KE]1 f | T, F | Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Formulation PFF* XF Pb*F* =PbP Pb’b*=PFF* Pb’F =PbP [KE]2 [KE]1 Xb’**

Context Kinematic Model Mechanical model Conclusion Machining forces and their displacement Error of the workpiece under load XF* PFF* XF {XE}2 Z X Y P [K]1 [K]2 [K]3 [K]4 [K]5 [K]6 {XE}1 [KE]2 [KE]1 f | T, F | Rigid link Pb*F* =PbP Pb’b*=PFF* Pb’F =PbP Clamping forces and their displacements Xb* Xb’ Correction Pb*b’ POb’ Pob* XO Initial baseplate locating under load Correction through locators Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Formulation Context Kinematic Model Mechanical model Conclusion Lagrangian Equation Z X Y 2 1 3 K3 6 K6 P ZP XP YP 5 K5 4 K4 K2 K1 Baseplate KE1 XE1 KE2 XE2 Part F T Machine/Pallet Reference U: Potential energy of the system T: Kinetic energy of the system W:Work done by the external forces qi: Generalized coordinates Work done by external force {ΔX,ΔY,ΔY}T: Linear displacement vector of point P {Δα,Δβ,Δγ}T: Angular displacement vector of point P {F}: Force vector {T}: Moment vector Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Formulation Context Kinematic Model Mechanical model Conclusion Lagrangian Equation Z X Y 2 1 3 K3 6 K6 P ZP XP YP 5 K5 4 K4 K2 K1 Baseplate KE1 XE1 KE2 XE2 Part F T Machine/Pallet Reference Potential Energy: U Locators Clamps Kinetic Energy: T (Lalanne et al. 1986) Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Locator model Context Kinematic Model Mechanical model Conclusion Locator Pin 68mm 20mm Screw-nut Wedge-slop locator Rotation of knob causes axial movement of locator Slope 1:2 Screw M6x1 Locator diameter: 20mm Length of locator: 68mm Sphere radius: 20mm Perforated plate Pitch = 40mm Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Formulation (Zero Friction) Context Kinematic Model Mechanical model Conclusion Locator axis Locator’s Stiffness Matrix 3 Initial position of the surface 2 Bending + Shear Compression 1 4 Final position of the surface Z Y X Deformed locator at the position having minimum potential energy Minimum energy (Menabrea’s theorem) Potential energy of locators Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Formulation Deformation of contact (Hertz contact theory) Context**

Kinematic Model Mechanical model Conclusion Deformation of contact (Hertz contact theory) Final position of contact surface after only locator deformation Final position of contact surface after locator and contact deformation Z Y X Zero contact deformation Including contact deformation Satisfactory results in 3-4 iterations Transition!! Iteration procedure Sajid Ullah BUTT, PhD Defense 5 July 2012

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**{∆X}= {∆XNew} +gain*({∆X} - {∆Xnew} )**

Formulation (Iterations procedure) Context Kinematic Model Mechanical model Conclusion {∆XNew} New stiffness matrices of each locator [K]i Kinetic energy and Work done Force vector on the ith locator {F}i 3D stiffness matrix of the locator body [KL] i [KC] i {F}i 3D stiffness matrix of the contact [K]i 3D equivalent stiffness matrix {F}i [KNew] i Potential energy calculations Stiffness matrix [K] , overall displacement vector {∆X} and natural frequencies of the system using Lagrangian Gain {∆X}= {∆XNew} +gain*({∆X} - {∆Xnew} ) Deformation of each locator{δ}i Z Y X Deformation and stiffness of ith locator body (δL, KL) i Deformation and stiffness of ith contact (δC, KC) i {δNew}i= {δC}i +{δL}i Overall stiffness of each locator and displacement vector of the workpiece ({∆Xnew} ) using inverse Plucker [KNew]i = {F}i /{δNew}iT {∆XNew}=[Plu]-1{δPlu} [K]i =[KNew]i {∆XNew} No Yes/STOP Final deformation/displacement vector and stiffness matrix of each locator and the fixturing system Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Case study Context Kinematic Model Mechanical model Conclusion {XE}2 Z X Y P [K]1 [K]2 [K]3 [K]4 [K]5 [K]6 {XE}1 [KE]2 [KE]1 f | T, F | Y X Z 1 2 3 4 5 6 P 70 110 14 100 8 60 23 21 22 10 120 40 O Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Case study (Input) Locator stiffness Baseplate (made of steel) Clamps**

Context Kinematic Model Mechanical model Conclusion Locator stiffness {XE}2 Z X Y P [K]1 [K]2 [K]3 [K]4 [K]5 [K]6 {XE}1 [KE]2 [KE]1 f | T, F | Baseplate (made of steel) Extracted from CAD model 68mm 20mm Clamps locators-baseplate contacting points Contacting points of clamps Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Case study (Input) Context Kinematic Model Mechanical model Conclusion Processed material Prosthesis material : M30NW (X4CrNiMoN21) ISO equivalent material: M3.2.C.AQ (Stainless steel Cast, Annealed quenched) Stainless steel 316LN (X2CrNiMoN18-13, Sandvik technical guide 2011) σ = 880MPa Cutting condition Tool : CoroMill 216 ball nose endmill of 3 mm in diameter (2 teeth) DOC, ap : mm, Feed per tooth, Fz : mm Cutting speed, Vc: 75 m/min, Spindle speed, N : RPM Machining forces Tangential force (Sandvik technical guide 2011) Repulsive forces (Pruvot, 1993) Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Case study (Results) Context Kinematic Model Mechanical model Conclusion Results without considering contact deformation Results with considering contact deformation Natural frequencies of the system Error compensation Tool Excitation frequency=837 rad/sec Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Convergence of displacement vector Context Kinematic Model Mechanical model Conclusion μRad, μm No of iterations % Error No of iterations Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Effect of gain on convergence Context Kinematic Model Mechanical model Conclusion Convergence of parameter slowest parameter ∆γ μRad No of iterations Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Rough contacts Context Kinematic Model Mechanical model Conclusion (Bahrami et al. 2005) Baseplate-locator equivalent RMS roughness = 0.8 E-6 m kC/kH RMS Roughness (m) 1N 10N 100N 1kN 6.6% decrease Comparison between ideal and rough surface contacts Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

4-2-2 locator configuration Context Kinematic Model Mechanical model Conclusion {XE}2 Z X Y P [K]1 [K]2 [K]3 [K]4 [K]5 [K]6 {XE}1 [KE]2 [KE]1 f [K]7 [K]8 | T, F | {XE}2 Z X Y P [K]1 [K]2 [K]3 [K]4 [K]5 [K]6 {XE}1 [KE]2 [KE]1 f | T, F | System stiffness Comparison between and 4-2-2 Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Conclusion and Perspectives**

Presentation layout Context Kinematic Model Mechanical model Conclusion Context Kinematic Model Mechanical Model Conclusion and Perspectives Context Positioning errors Compensation Objectives Workpiece repositioning through locators All the elements rigid Proposed fixturing system Analytical formulation Large displacements and Homogeneous Transformation Matrices Deformation of elastic elements and rigid body displacement of part on fixture under load Deformation of locators and contacts Lagrangian formulation Negligible friction, Small Displacements Convergence of non-linear contact deformation Work realized Conclusion Future work Case Study Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Conclusion Context Kinematic Model Mechanical model Conclusion High quality baseplate is introduced Compensation is performed through advancement of locators (kinematic model) Deformation of each locator is calculated and its contact with baseplate under load (mechanical model) Also its resultant rigid body displacement of the workpiece is calculated (positioning error) Compensation is performed using kinematic model Column Kinematics defects Tool Part Baseplate Part Baseplate 6 DOF part repositioning Part Locators placement Pallet Deformation due to forces Geometric/form defects Base Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Conclusion Context Kinematic Model Mechanical model Conclusion 6 DOF positioning using kinematic model 3-2-1 locating configuration is used All elements are considered rigid Error compensation is performed through the axial translation of 6-locators using HTM and LD Validated on a case study of repositioning the hip prosthesis Sensitivity analysis are carried out Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Conclusion Context Kinematic Model Mechanical model Conclusion Mechanical modeling of the fixturing system The analytical model is developed Deformation of locators under load is calculated Workpiece-baseplate assembly is designed to be rigid Locators, clamps and locator-baseplate contacts are assumed deformable Small displacement hypothesis are used Lagrangian formulation is used to calculate the mass & stiffness matrices and workpiece displacement vector Non-linear behavior of locator-baseplate contact is linearized Demonstrated on a case study of locating configuration and compared with configuration of locators Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Conclusion Context Kinematic Model Mechanical model Conclusion Analytical modeling gives very quick result as compared to numerical modeling The proposed mechanical model can easily be applied to more complex problems with multiple loads, different orientations and stiffness of locators and clamps The proposed fixturing system allows precise positioning of the workpiece at each workstation without the need of 4 or 5 axis machines or modifying the existing workstations Reduce dimensional errors, machining allowances and thus the material removal by uniformly centering the rough part to the required part Consequently, it reduces the material waste Large parts could also be repositioned during assembling Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Sajid Ullah BUTT, PhD Defense 5 July 2012**

Limitations and Future work Context Kinematic Model Mechanical model Conclusion The positioning error due to heat/temperature change can be introduced Construction of the fixturing system for validating the analytical model. We could not construct the model because of time and cost associated with precise part production The mechanical model calculates the deformation of locators as the result of an instantaneous force at a point. The model should be developed to simulate the whole tool path Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Scientific activities**

Poster Presentation “Conception, Modélisation et Réalisation d'un montage modulaire rapide destine a la Fabrication Mécanique” J2A Arts et Métiers ParisTech, 8th-9th June 2010 (Poster Presentation) National Colloquium S. U. Butt, J. F. Antoine, P. Martin, “Mechanical model for control of 6 DOF repositioning system”, 12th National Colloquium, AIP Primeca, Mont-Dore 29th March to 1st April 2011 Scientific Publications S. U. Butt, J. F. Antoine, P. Martin, “An analytical model for the repositioning of 6 DOF repositioning system”, Journal of Mechanics and Industry (Accepted June 2012 ) S. U. Butt, J. F. Antoine, P. Martin, “An analytical stiffness model for spherical rough contacts”, Asian International Journal of Science and Technology in Production and Manufacturing Engineering (Submitted June 2012) Sajid Ullah BUTT, PhD Defense 5 July 2012

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**Thanks for your attention**

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**Measuring locator positioning accuracy**

Measuring the accuracy of the locator movement Designed slope 1:2 Obtained av. slope:

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**Measuring locator positioning accuracy**

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**Backlash in the advancement**

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**Position of the workpiece**

Z Y X O β γ Y0 XP Z0=Z1 X0 Y2=YP X1=X2 Z2 ZP Y1 P(xP, yP, zP) α Z0=Z1 P YPR Transformation (Yaw, Pitch,Roll)

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**Baseplate position in machine**

X Y Z 3 1 2 4 5 6 u w v b

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